![]() Remarkably, the above discussion of the quantum treatment can be generalized to purely classical scenarios. Finally, we discuss the physical meaning of the conditional thermal states in the second law of thermodynamics based on its ergotropy in Section 6 before our conclusions in Section 7.Ĥ. Then, we introduce the formulation of classical ergotropy in Section 4, and discuss the geometric quantum mechanics approach to ergotropies in Section 5. In Section 2, we review quantum ergotropy in terms of relative entropies and its relation to the quantum coherence in Section 3. This analysis further cements ergotropy as one of the salient pillars of quantum thermodynamics. Hence, by expressing the quantum ergotropy as a difference of relative entropies, we are able to (i) generalize the notion to classical scenarios, and to (ii) elucidate the thermodynamics of projective measurements. Here, we identify the distinct contributions to the thermodynamic cost of projective measurements by separating out the coherent and incoherent ergotropies, and the population mismatch in the conditional statistics. Hence, the work probability distribution is entirely determined by the statistics conditioned on the initial energy. In contrast to the two-time measurement approach, no projective measurement is taken at the end of the process. In this paradigm, work is determined by first measuring the energy of the system, and then letting it evolve under time-dependent dynamics. With this, it becomes particularly transparent to characterize the one-time measurement approach to quantum work. ![]() Exploiting this approach, we define the geometric relative entropy. In a second part of the analysis, we turn to a unified framework, namely geometric quantum mechanics. In this paper, given that the quantum ergotropy can be written as the difference of quantum and classical relative entropies (the Kullback–Leibler divergence of the eigenvalue distributions), we define a classical ergotropy, which quantifies the maximal amount of work that can be extracted from inhomogeneities on the energy surfaces, which have been shown to be analogous to quantum coherences. Note that not all passive states can be reached by unitary operations, in particular, including the completely passive state. ![]() This is due to the fact that the ergotropy is determined by a maximum over all unitaries that can act upon the system. ![]() However, if the quantum system is not in contact with a heat reservoir, computing the quantum ergotropy is far from trivial. ![]() In particular, when assessing the thermodynamic value of genuine quantum properties, such as squeezed and nonequilibrium reservoirs, coherence, or quantum correlations, it has proven powerful. The quantum ergotropy plays a prominent role in quantum thermodynamics. Gibbs states are then called completely passive. In simple terms, a passive state is diagonal in the energy basis, and its eigenstates are ordered in descending magnitude of its eigenvalues. In particular, the quantum ergotropy quantifies the amount of energy that is stored in active quantum states, and which can be extracted by making the state passive. Generalizing this notion to physical systems, quantum ergotropy was then coined to denote the maximal amount of work that can be extracted by isentropic transformations. According to its definition, the adjective ergotropic refers to the physiological mechanisms of a nervous system to favor an organism’s capacity to expend energy. ![]()
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